Talk:Circle packing theorem

Disjoint?

What does disjoint mean in this context? Thanks. Saintrain (talk) 00:23, 14 October 2008 (UTC)[reply]

Not having any shared points. —David Eppstein (talk) 00:27, 14 October 2008 (UTC)[reply]

So "A packing of circles is a collection of circles whose interiors are disjoint." means none of the circles overlap? Is there any other meaning than that they don't overlap? Saintrain (talk) 12:34, 14 October 2008 (UTC)[reply]

It is not completely clear if your "the circles don't overlap" forbids the boundaries to have shared points or not. So, instead of "the interiors are disjoint" you could say "the interiors don't overlap", but stressing the interiors is important. GaborPete (talk) 05:18, 16 August 2009 (UTC)[reply]

Origami Circle Packing

I do not think the use of Circle Packing in Origami Design refers to the Circle Packing Theorem. For instance, in http://arxiv.org/pdf/1008.1224v2 Circle Packing is said to be the problem of packing some number of circles of known radius into the unit square. This is the same use of the term "Circle Packing" as in http://en.wikipedia.org/wiki/Circle_packing . The term "Circle Packing" is ambiguous because it refers to both the 2D analog of the Sphere Packing problem (which is what I think Origami Design uses), and the Circle Packing as described in this article. — Preceding unsigned comment added by 131.229.145.70 (talk) 18:30, 2 December 2011 (UTC)[reply]

Coin graphs necessarily connected?

The proposition (in the lead) seems so self-evidently wrong that I had to stop myself from editing it out. Clearly a graph as simple as 2K₁ has a coin realization in the disk pair

x² + y² ≤ 1

and

(x - 3)² + y² ≤ 1 .

Am I missing something? Unless somebody comes on line shortly to educate me, I'm going to delete the offending proposition.—PaulTanenbaum (talk) 18:14, 22 March 2013 (UTC)[reply]

Yes, it seems wrong as stated — we need to either delete the word "connected" or add an assumption that the union of the circles is a connected set. I think deleting the word is the better choice. —David Eppstein (talk) 18:55, 22 March 2013 (UTC)[reply]

It actually appears to be correct, because it defines "A circle packing is a connected collection of circles [...]". I take this to mean that the union of the circles is a connected set, as David suggested. By this definition your example is not a circle packing. It seems to me like the theorem will still hold in both directions even if the connectedness requirement is removed (just take the connected components and put them far enough apart in the plane that their circle packings don't overlap) but I prefer not to mess with it. Dcoetzee 15:58, 24 March 2013 (UTC)[reply]

I make no claim about these particular coins (i.e., in the context of the circle packing theorem). My objection is to the sentence as written; it asserts that "coin graphs are always connected," [emphasis added] and that is false, as my example above shows.

Is it as simple as prefixing one modifier: "such coin graphs are always connected"?—PaulTanenbaum (talk) 19:41, 25 March 2013 (UTC)[reply]

Koebe-Andre'ev-Thurston vs. Circle Packing

The Koebe-Andre'ev-Thurston Theorem and the Circle Packing Theorem are not the same. The Circle Packing Theorem deals with tangent neighboring circles only, while KAT allows for circles to overlap with angles up to pi/2 (subject to some mild conditions). KAT is a generalization of the Circle Packing Theorem.— Preceding unsigned comment added by 204.111.113.34 (talk) 15:36, 25 April 2022 (UTC)[reply]